The present invention relates to a data processing method and a data processing apparatus for processing a series of numerical data on a target system (device, analyzed data, or the like).
Various models have been proposed as techniques for smoothing or predicting a series of numerical data on a target system. In addition, aiming at grasping the state of a target system, differential processes such as first differential and second differential of data are frequently used so as to detect an extreme value (maximum value or minimum value) or an inflection point, which is a changing point in the data. In particular, in the case of time series data such as measured data containing noise and the like, a technical issue is to perform a data smoothing process and a differential process with high precision to detect a changing point in a target system, and to control the target system.
Conventional techniques to perform data smoothing and data prediction include a curve fitting method, a moving average method, and the like, as described in K. Takahashi, “Inside Data Processing”, Journal of Surface Analysis Vol. 7, No. 1, 2000, p. 68-p. 77. The curve fitting method includes a polynomial curve-fitting method (Savitzky-Golay method), as described in JP-A-2000-228397, and in addition, a digital filter includes a Butterworth low-pass filter. Furthermore, the moving average method includes an exponential smoothing method, and the like, as described in A. C. Harvey, “TIME SERIES MODELS”, translated by N. Kunitomo & T. Yamamoto, University of Tokyo Press, 1985, p. 173. Although “TIME SERIES MODELS” discloses a single exponential smoothing method (with one smoothing parameter), a double exponential smoothing method (with two smoothing parameters) is also used in economics-related fields such as supply and demand estimation.
For conventional techniques to perform a data differential process, a finite difference method is frequently used, as described in “Fluid Calculation and Finite Difference Method” written and edited by K. Kuwahara & T. Kawamura, Asakura Publishing, 2005, p. 1. In addition, it is the case that a polynomial curve-fitting method (Savitzky-Golay method) is used therefor, as described in JP-A-2000-228397. Furthermore, as an example in which time series data such as measured data containing noise and the like is subjected to a data smoothing process, a changing point of data is detected through a first differential process and a second differential process, and a target system is controlled, JP-A-61-53728 discloses a method in a plasma etching processing apparatus that includes subjecting spectral intensity signal data based on plasma emission to a data smoothing process by a moving average process and determining the end point of etching process with a first differential value and a second differential value.
As described in JP-A-2000-228397, when the data differential process is performed by the finite difference method and if the data smoothing process is not sufficient, the output result of the first differential process contains much noise, resulting in data being not smooth and having a low S/N ratio (ratio of signal/noise). When the data differential process is performed again by the finite difference method using the above data, the output result of the second differential process contains still more noise, resulting in data being not smooth and having a further low S/N ratio, which is problematic.
When difference intervals are increased, the output result of the first differential process and the output result of the second differential process are obtained as smooth data and have an increased S/N ratio (note that there is an optimal value for the difference interval), but in the case of the time series data, in particular, there is a problem that the amount of data delay caused by the data differential process increases. In addition, in a low-pass filter, or the like, when a cut-off frequency is decreased, the output data of the first differential process and the output data of the second differential process are obtained as smooth data and have an increased S/N ratio (note that there is an optimal value for the cut-off frequency), but as with the above, there is a problem in that the amount of data delay caused by the data differential process increases.
Furthermore, in the case where the data differential process is performed using a polynomial curve-fitting method (Savitzky-Golay method), a plurality of pieces of data are generally needed and a differential value is derived at a point in time of the middle piece of the data. For this reason, there is a problem with the sequential data processing that a time delay occurs in principal by at least a time difference between a point in time of the latest piece of the data and the point in time of the middle piece of the data.
Furthermore, in the data smoothing process, errors tend to develop in general during a certain period of time immediately after the start of the data processing. For example, in the case of the time series data, the data period is short and a sampling time interval is long, and thus if the number of pieces of the data is small, a ratio of the period during which the error is large with respect to the whole data period is large. In addition, there is a problem that performing the data differential process in the period during which the error is large has little reliability.
Furthermore, there is a method disclosed in JP-A-61-53728 as an end point determining method in the plasma etching in the plasma etching processing apparatus, but the end point determining method in the plasma etching disclosed in JP-A-61-53728 involves the following two problems.
Current semiconductor devices each have a high-step structure such as a Fin Field Effect Transistor (Fin FET) structure due to higher performance and integration. In addition, in normal plasma etching, microloading occurs, which is the difference in etching performance between sparse patterns and dense patterns. Furthermore, a thickness of film to be etched is not uniform across the surface of a wafer.
For these reasons, for example, there is the case where time series data on spectral intensity signal data based on plasma emission used for determining the end point of plasma etching changes in two stages. Hence, in the case where the time series data on the spectral intensity signal data based on the plasma emission changes in two stages, and the end point of the plasma etching is determined at the second change, data processing cannot track the second change and the end point of the plasma etching cannot be detected because the first change and the second change occur in a short time. Note that, here, a point in time at which the time series data on the spectral intensity signal data based on the plasma emission changes is defined as the end point of the plasma etching.
Furthermore, for example, in a plasma etching the period of which is short with respect to retardation times of a first differential value and a second differential value, in case where the end point of plasma etching is determined by the change of the time series data on the spectral intensity signal data based on the plasma emission, data processing to calculate the first differential value or the second differential value cannot track a change, to be the end point of plasma etching, in time series data on spectral intensity signal data based on plasma emission. That is, the responsiveness of detecting the end point of plasma etching process is insufficient, which is the first problem.
Next, mask patterns for plasma etching are roughly divided into groove patterns and hole patterns. In addition, an aperture ratio of a wafer of a hole pattern is normally less than an aperture ratio of a wafer of a groove pattern, and there is even a case where the aperture ratio is less than 1%. Furthermore, the spectral intensity of plasma emission is reduced as the aperture ratio becomes small. For this reason, for example, in the case of a wafer of an aperture ratio of less than 1%, it is difficult to detect the end point of plasma etching process since a change in the time series data on the spectral intensity signal data based on the plasma emission is too small. That is, a low S/N ratio cannot be supported, which is the second problem. Note that, here, the aperture ratio is a ratio of an area to be etched to the entire area of the wafer.
For the foregoing reasons, in the case of detecting a changing point in a target system with first differential data or second differential data and controlling the target system, there is a problem that the precision of the control is insufficient due to a low S/N ratio and a time delay, or the like.